3.72.46 \(\int \frac {e^{\frac {-4+63 x+16 x^2+e^4 (-4 x-x^2)}{x}} (4+16 x^2-e^4 x^2)}{x^2} \, dx\)

Optimal. Leaf size=18 \[ e^{\left (16-e^4-\frac {1}{x}\right ) (4+x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.47, antiderivative size = 28, normalized size of antiderivative = 1.56, number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6, 6706} \begin {gather*} e^{-\frac {-16 x^2+e^4 \left (x^2+4 x\right )-63 x+4}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-4 + 63*x + 16*x^2 + E^4*(-4*x - x^2))/x)*(4 + 16*x^2 - E^4*x^2))/x^2,x]

[Out]

E^(-((4 - 63*x - 16*x^2 + E^4*(4*x + x^2))/x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-4+63 x+16 x^2+e^4 \left (-4 x-x^2\right )}{x}} \left (4+\left (16-e^4\right ) x^2\right )}{x^2} \, dx\\ &=e^{-\frac {4-63 x-16 x^2+e^4 \left (4 x+x^2\right )}{x}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 22, normalized size = 1.22 \begin {gather*} e^{63-4 e^4-\frac {4}{x}-\left (-16+e^4\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-4 + 63*x + 16*x^2 + E^4*(-4*x - x^2))/x)*(4 + 16*x^2 - E^4*x^2))/x^2,x]

[Out]

E^(63 - 4*E^4 - 4/x - (-16 + E^4)*x)

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 26, normalized size = 1.44 \begin {gather*} e^{\left (\frac {16 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{4} + 63 \, x - 4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(4)+16*x^2+4)*exp(((-x^2-4*x)*exp(4)+16*x^2+63*x-4)/x)/x^2,x, algorithm="fricas")

[Out]

e^((16*x^2 - (x^2 + 4*x)*e^4 + 63*x - 4)/x)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 20, normalized size = 1.11 \begin {gather*} e^{\left (-x e^{4} + 16 \, x - \frac {4}{x} - 4 \, e^{4} + 63\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(4)+16*x^2+4)*exp(((-x^2-4*x)*exp(4)+16*x^2+63*x-4)/x)/x^2,x, algorithm="giac")

[Out]

e^(-x*e^4 + 16*x - 4/x - 4*e^4 + 63)

________________________________________________________________________________________

maple [A]  time = 0.13, size = 19, normalized size = 1.06

method	result	size
risch	\({\mathrm e}^{-\frac {\left (4+x \right ) \left (x \,{\mathrm e}^{4}-16 x +1\right )}{x}}\)	\(19\)
gosper	\({\mathrm e}^{-\frac {x^{2} {\mathrm e}^{4}+4 x \,{\mathrm e}^{4}-16 x^{2}-63 x +4}{x}}\)	\(28\)
norman	\({\mathrm e}^{\frac {\left (-x^{2}-4 x \right ) {\mathrm e}^{4}+16 x^{2}+63 x -4}{x}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2*exp(4)+16*x^2+4)*exp(((-x^2-4*x)*exp(4)+16*x^2+63*x-4)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(-(4+x)*(x*exp(4)-16*x+1)/x)

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 20, normalized size = 1.11 \begin {gather*} e^{\left (-x e^{4} + 16 \, x - \frac {4}{x} - 4 \, e^{4} + 63\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(4)+16*x^2+4)*exp(((-x^2-4*x)*exp(4)+16*x^2+63*x-4)/x)/x^2,x, algorithm="maxima")

[Out]

e^(-x*e^4 + 16*x - 4/x - 4*e^4 + 63)

________________________________________________________________________________________

mupad [B]  time = 4.20, size = 24, normalized size = 1.33 \begin {gather*} {\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{16\,x}\,{\mathrm {e}}^{63}\,{\mathrm {e}}^{-\frac {4}{x}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((63*x - exp(4)*(4*x + x^2) + 16*x^2 - 4)/x)*(16*x^2 - x^2*exp(4) + 4))/x^2,x)

[Out]

exp(-4*exp(4))*exp(16*x)*exp(63)*exp(-4/x)*exp(-x*exp(4))

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 24, normalized size = 1.33 \begin {gather*} e^{\frac {16 x^{2} + 63 x + \left (- x^{2} - 4 x\right ) e^{4} - 4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2*exp(4)+16*x**2+4)*exp(((-x**2-4*x)*exp(4)+16*x**2+63*x-4)/x)/x**2,x)

[Out]

exp((16*x**2 + 63*x + (-x**2 - 4*x)*exp(4) - 4)/x)

________________________________________________________________________________________